Remember -- use your compass
and straightedge only!

Bisect a line segment
Note: This construction is also the construction for Perpendicular Bisector of a Segment.
Given: AB (a line segment)
Construction: bisect AB.
Remember that "bisect" means to form two congruent segments.
 segmentAB
This construction will also establish that the bisector is perpendicular to the given line segment.

STEPS:
1. Place your compass point on A and stretch the compass MORE THAN half way to point B (you may also stretch to point B).

2. With this length, swing a large arc that will go above and below AB.

3. Without changing the span on the compass, place the compass point on B and swing the arc again. The two arcs need to be extended sufficiently so they will intersect in two locations (one above the segment and one below the segment).

4. Using your straightedge, connect the two points of intersection with a line or segment to locate point C which bisects the segment.

bisectsegment
bisectmath

hintgal
You may also see this construction done where only small portions of the arcs are shown both above and below the segment. The larger arcs, as shown here, are often easier to remember (since they form a "crayfish" looking creature) and they visually reinforce the circle concept needed for the proof of this construction.

See Contruct: Divide a Segment to see what happens when you bisect a segment,
then bisect the parts, and continue this bisecting process.

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Proof of Construction: Label the points of intersection of the arcs with the letters D and E. Draw segments segmentsbisectlist These four segments will be congruent as they are the radii of two congruent circles. We now know that point D is equidistant from points A and B. And we know that point E are also equidistant from points A and B. Theorem: "The points on a perpendicular bisector of a line segment are equidistant from the endpoints of the line segment." Therefore, is the perpendicular bisector of . (And bisects .)
bisectsegmentcircles

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Alternative Proof Using Congruent Triangles:

bewaresm
The congruent triangle proof of this simple construction is quite lengthy.
Here we go!

Proof of Construction: Label the points of intersection of the arcs with the letters D and E. Draw segments segmentsbisectlist These four segments will be congruent as they are the radii of two congruent circles. We can now show that there are four congruent triangles in this diagram giving us the congruent segments that will prove that the bisecting has occurred. This proof will use two sets of congruent triangles.

bisectsegmentcircles

First congruentS Triangles:
bisectshade1

With segswe know ΔDAE and ΔDBE are congruent by SSS (they share sideDE2).

Now, ∠BDE congruentS∠ADE since corresponding parts of congruentS Δs are congruentS (CPCTC).

 

Second congruentS Triangles:
bisectingshade2
With ∠BDE congruentS∠ADE, and the shared side ac, ΔADC and ΔBDC are congruent by SAS.

Now, segbisby CPCTC.

We now have DE2 bisecting ab2 since two congruent segments were formed.

 

Perpendicular:
bisectingshade2
We also have ∠DCA congruentS∠DCB by CPCTC.

These ∠s form a linear pair making them supplementary.
m∠DCA + m∠DCB = 180 (definition of supplementary).
By substitution, m∠DCA +
m∠DCB = 180, making
m∠DCA = 90.

∠DCA is a right angle by definition, making deabperpsince ⊥lines form rt.∠s.


hintgal
Notice that after this construction, the 4 sides of quadrilateral ADBE are congruent making ADBE a rhombus. We know that the diagonals of a rhombus bisect each other and are also perpendicular, supporting our construction.



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